Illinois Journal of Mathematics

Another approach to biting convergence of Jacobians

Luigi Greco, Tadeusz Iwaniec, and Uma Subramanian

Full-text: Open access

Abstract

We give new proof of the theorem of K. Zhang [Z] on biting convergence of Jacobian determinants for mappings of Sobolev class ${\mathscr W}^{1,n}(\Omega,\mathbb{R}^n)$. The novelty of our approach is in using ${\mathscr W}^{1,p}$-estimates with the exponents $1\leqslant p \lt n$, as developed in [IS1], [IL], [I1]. These rather strong estimates compensate for the lack of equi-integrability. The remaining arguments are fairly elementary. In particular, we are able to dispense with both the Chacon biting lemma and the Dunford-Pettis criterion for weak convergence in ${\mathscr L}^1(\Omega)$. We extend the result to the so-called Grand Sobolev setting.

Biting convergence of Jacobians for mappings whose cofactor matrices are bounded in ${\mathscr L}^{\frac n{n-1}}(\mathbb{R}^n)$ is also obtained. Possible generalizations to the wedge products of differential forms are discussed.

Article information

Source
Illinois J. Math., Volume 47, Number 3 (2003), 815-830.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138195

Digital Object Identifier
doi:10.1215/ijm/1258138195

Mathematical Reviews number (MathSciNet)
MR2007238

Zentralblatt MATH identifier
1060.46023

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 30C65: Quasiconformal mappings in $R^n$ , other generalizations 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 49J45: Methods involving semicontinuity and convergence; relaxation

Citation

Greco, Luigi; Iwaniec, Tadeusz; Subramanian, Uma. Another approach to biting convergence of Jacobians. Illinois J. Math. 47 (2003), no. 3, 815--830. doi:10.1215/ijm/1258138195. https://projecteuclid.org/euclid.ijm/1258138195


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